A numerical method based on a bilinear pseudo-spectral method to solve the convection-diffusion optimal control problems

In this paper, we consider the bilinear pseudo-spectral method for solving convection-diffusion optimal control problems (OCPs). First, we convert the optimal control problem to a partial differential equation (PDE) system including the state equation of the original problem and the adjoint equation which must be solved. Next, we approximate the coupled system by a bilinear pseudo-spectral method based on Chebyshev polynomials and obtain a coupled Sylvester system and then use some iterative and direct methods to solve it. We used bilinear pseudo-spectral method to have simplicity in implementation and Chebyshev polynomials to have accuracy and stability. Robustness and accuracy of the method are verified by solving some numerical experiments.